For any interval \(I \subset \mathbb {R}_{+}\) , we shall denote by \(I, \mathbb {R})\)  the space of real-valued càdlàg functions on \(I\) , equipped with the Skorokhod \(J_{1}\)  topology (see Chapter VI in [1]} for background). For any \(\alpha \in (1,2]\) , we shall denote by \(X^{\alpha }\)  the \(\alpha \) -stable Lévy process with no negative jump, whose law is characterised by the Laplace transform \(\mathbb {E}[\exp (-q X^{\alpha }_{t})] = \exp (t q^{\alpha })\)  for every \(q,t > 0\) ; note that \(X^{2}\)  has the law of \(\sqrt{2}\)  times a standard Brownian motion. Similarly to the discrete setting, we can make sense of the law of \((X^{\alpha }_{t}-ct ; 0 \leqslant t \leqslant 1)\)  conditioned on the event \(\lbrace X^{\alpha }_{1} - c=0\rbrace \)  with \(c \in \mathbb {R}\) . We refer to e.g. [2]}, Chapter XIII, and especially Sec. 3 there for details (which focuses in the case \(c=0\) , but the arguments extend readily).
